To answer your question, yes it is possible.
Is it a good idea? Probably not. The computer would have to be running and booted (although not to the OS) the entire time. You'd also have a greater risk that it would die (the logic board or memory could go out and it would stop working even though the HDD was fine).
I'd suggest (as have others) that you sell the MacBook and use the money to buy an external drive. There are lots of options. USB, FireWire, Thunderbolt, TimeCapsule, etc.
You could also give it as a gift and become someone's favorite Aunt/Uncle/cousin/etc..